2025/09/23 by Dali, Houcine Ben
#05E05 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2509.18625
We study a Jack analog ∇(p,q) of the super nabla operator recently introduced by Bergeron, Haglund, Iraci and Romero for Macdonald polynomials. We prove that ∇(p,q) has a differential expression in the power-sum basis given in terms of Chapuy--Dołe\kga and Nazarov--Sklyanin operators. This result is obtained from a more general formula for the operator G(p,q) encoding the structure coefficients of Jack characters, from which ∇(p,q) is obtained by taking the top homogeneous part. A key step of the proof involves establishing that Chapuy--Dołe\kga operators together with a dehomogenized version of Nazarov--Sklyanin operators have a Heisenberg algebra structure. The proof also uses a characterization of the operator G(p,q) with a family of differential equations, recently established by the author.